Your statement is surely true in the following two cases:

1. $\dim X=2$. In this case something stronger is true; in fact, if $X$ is a surface with only rational singularities and $\tilde{X}$ is a resolution of singularities, then $\pi_1(\tilde{X})=\pi_1(X)$. This because the resolution of a rational $2$-dimensional singularity is alwais given by a bunch of rational curves, in particular the exceptional locus is simply connected, so you can conclude by Seifert-Van Kampen theorem. Thi argument is used for instance in the paper of R. Barlow

"A symply connected surface of general type with $p_g=0$", Invent. Math.79.

1. $\dim X$ is arbitrary and all singularities of $X$ are isolated cyclic quotient singularities. In fact, by a result of Fujiki, an isolated cyclic quotient singularity can always be resolved by a normal crossing divisor whose components are smooth rational varieties, so you can apply again Seifert-Van Kampen theorem.

This argument is used in the paper of Bauer, Catanese, Pignatelli, Grunewald

Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

in order to compute the fundamental group of a quotient of the form $(C_1 \times C_2 \times \cdots \times C_n)/G$, where the $C_i$ are smooth curves (see in particular Remark 2.4).

I do not know whether the result is true in full generality. In fact, given a finite group $G$ acting on a simply connected manifold $V$, by a result of Armstrong (Proc. Amer. Math. Soc. 84) one has

$\pi_1(V/G)=G/E$,

where $E$ is the subgroup of elements having fixed points.

In your case $V=\mathbb{C}^n$, and if $\mathbb{C}^n/G$ is simply connected then you can apply Seifert-Van Kampen again. So the question is whether any cyclic group $\langle g \rangle$ acting on $\mathbb{C}^n$ has a fixed point. If $g$ is a polynomial automorphism then the answer is yes, but in general this is an open problem for $n \geq 3$ (for $n=2$ it is true since any finite group acting on $\mathbb{C}^2$ is linearizable by a result of Suzuki).

You can find more on this topic (and several related references) in the paper by Kraft and Schwarz "Finite automorphisms of affine $n$-spaces".

Your statement is surely true in the following two cases:

1. $\dim X=2$. In this case something stronger is true; in fact, if $X$ is a surface with only rational singularities and $\tilde{X}$ is a resolution of singularities, then $\pi_1(\tilde{X})=\pi_1(X)$. This because the resolution of a rational $2$-dimensional singularity is alwais given by a bunch of rational curves, in particular the exceptional locus is simply connected, so you can conclude by Seifert-Van Kampen theorem. Thi argument is used for instance in the paper of R. Barlow

"A symply connected surface of general type with $p_g=0$", Invent. Math.79.

1. $\dim X$ is arbitrary and all singularities of $X$ are isolated cyclic quotient singularities. In fact, by a result of Fujiki, an isolated cyclic quotient singularity can always be resolved by a normal crossing divisor whose components are smooth rational varieties, so you can apply again Seifert-Van Kampen theorem.

This argument is used in the paper of Bauer, Catanese, Pignatelli, Grunewald

Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

in order to compute the fundamental group of a quotient of the form $(C_1 \times C_2 \times \cdots \times C_n)/G$, where the $C_i$ are smooth curves (see in particular Remark 2.4).

I do not know whether the result is true in full generality. In fact, given a finite group $G$ acting on a simply connected manifold $V$, by a result of Armstrong (Proc. Amer. Math. Soc. 84) one has

$\pi_1(V/G)=G/E$,

where $E$ is the subgroup of elements having fixed points.

In your case $V=\mathbb{C}^n$, and if $\mathbb{C}^n/G$ is simply connected, i.e. if $G=E$, then you can apply Seifert-Van Kampen again. Therefore it seems to me that your question is related to the following one:

Does any cyclic group $\langle g \rangle$ acting on $\mathbb{C}^n$ have a fixed point?

If $g$ is a polynomial automorphism then the answer is yes, but in general this is an open problem for $n \geq 3$ (for $n=2$ it is true since any finite group acting on $\mathbb{C}^2$ is linearizable by a result of Suzuki).

You can find more on this topic (and several related references) in the paper by Kraft and Schwarz "Finite automorphisms of affine $n$-spaces".